A Guide to Indices (Powers)
Indices (also known as powers or exponents) are a fundamental part of maths that provide a simple way to write and work with repeated multiplication. From calculating computer storage in gigabytes ($10^9$ bytes!) to understanding compound interest, mastering indices is key to tackling complex problems with confidence.
What are Indices? The Core Concepts
Positive, Zero & Negative Indices
An index tells you how many times to multiply the base by itself. But what happens when the index isn’t a positive whole number?
- Positive Index: $2^3 = 2 \times 2 \times 2 = 8$.
- Zero Index: Any non-zero number to the power of zero is 1. Example: $2^0 = 1$.
- Negative Index: This means “find the reciprocal”. So, $a^{-n} = \frac{1}{a^n}$. Example: $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$.
The Laws of Indices
These rules are essential shortcuts for simplifying expressions. They only work when the base is the same!
1. Multiplication Law
When multiplying powers, you add the indices.
$a^m \times a^n = a^{m+n}$
Example: $2^3 \times 2^4 = 2^7$
2. Division Law
When dividing powers, you subtract the indices.
$a^m \div a^n = a^{m-n}$
Example: $5^6 \div 5^2 = 5^4$
3. Power of a Power Law
When raising a power to another power, you multiply the indices.
$(a^m)^n = a^{mn}$
Example: $(3^2)^3 = 3^6$
Worked Examples
Example 1: Negative Index
Calculate $5^{-2}$.
A negative index means we take the reciprocal and make the index positive.
$5^{-2} = \frac{1}{5^2} = \frac{1}{25}$
Example 2: Multiplication Law
Simplify $3^5 \times 3^4$.
The bases are the same, so we add the indices.
$3^5 \times 3^4 = 3^{5+4} = 3^9$
Example 3: Division Law
Simplify $7^8 \div 7^3$.
The bases are the same, so we subtract the indices.
$7^8 \div 7^3 = 7^{8-3} = 7^5$
Example 4: Combining Laws
Simplify $\frac{4^2 \times 4^7}{4^3}$.
- Simplify the numerator (multiplication law):
$4^2 \times 4^7 = 4^{2+7} = 4^9$. - The expression is now $\frac{4^9}{4^3}$.
- Simplify the fraction (division law):
$4^{9-3} = 4^6$.
Tutor Insights
🤔 Common Misunderstandings
- Negative Powers vs. Negative Numbers: Thinking $2^{-3}$ gives a negative answer. It doesn’t! It means $\frac{1}{2^3}$, which is a positive fraction.
- The Zero Power: Forgetting that any non-zero number to the power of zero is always 1.
- Mixing up the Laws: Multiplying powers instead of adding them (e.g., for $a^m \times a^n$), or adding instead of multiplying (e.g., for $(a^m)^n$).
📝 Common Exam Mistakes
- Applying laws to different bases: The rules do not work for $2^3 \times 3^2$. You must calculate each part separately.
- Simple calculation errors: Mistakes in basic arithmetic, even when the index rule is applied correctly.
- Not simplifying fully: Leaving an answer as $2^5$ when the question asks for a final numerical value (32).
Practice Questions
- Calculate: $4^{-2}$.
- Simplify, leaving your answer in index form: $5^7 \times 5^2$.
- Simplify, leaving your answer in index form: $11^9 \div 11^4$.
- Calculate the value of: $2^3 \times 2^{-1}$.
- Simplify, leaving your answer in index form: $(4^{-2})^3 \times 4^8$.
Show Answers
- Working: $4^{-2} = \frac{1}{4^2} = \frac{1}{16}$.
- Working: $5^{7+2} = 5^9$.
- Working: $11^{9-4} = 11^5$.
- Working: $2^{3+(-1)} = 2^2 = 4$.
- Working: $4^{-6} \times 4^8 = 4^{-6+8} = 4^2$.
FAQs
Q: What’s the difference between an index and an exponent?
A: They mean the same thing! “Index” is more common in the UK, while “exponent” is more common in the US. They both refer to the small number at the top.
Q: What if the base is negative?
A: Be careful with the sign of your answer! If a negative base is raised to an even power, the answer is positive (e.g., $(-3)^2 = 9$). If it’s raised to an odd power, the answer is negative (e.g., $(-3)^3 = -27$).
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